Sum and Difference Identities: Step-by-Step Guide
Sum and difference identities let you find the trig functions of a + b or a - b from the trig functions of a and b. sin(a +/- b) = sin a cos b +/- cos a sin b. cos(a +/- b) = cos a cos b -/+ sin a sin b. tan(a +/- b) = (tan a +/- tan b) / (1 -/+ tan a tan b).
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Split the angle into known angles. Write your angle as a sum or difference of common angles like pi/6, pi/4, and pi/3.\[\theta = a \pm b\]
- Choose the identity. Pick the sin, cos, or tan formula that matches the function and the plus or minus.\[\sin(a + b) = \sin a \cos b + \cos a \sin b\]
- Look up the values. Find the trig values for each part from the unit circle.\[\sin\tfrac{\pi}{4} = \tfrac{\sqrt{2}}{2},\;\cos\tfrac{\pi}{6} = \tfrac{\sqrt{3}}{2}\]
- Substitute and simplify. Multiply, combine, and simplify any roots.\[\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{4}\]
- Check with a calculator. Compare with a calculator value of the original expression.\[\text{decimal check}\]
Worked example
Find sin(7*pi/12) exactly using the sum identity.
Step 1: 7 pi / 12 = pi / 3 + pi / 4.
Step 2: Use sin(a + b) = sin a cos b + cos a sin b.
Step 3: sin(pi/3) = root 3 / 2, cos(pi/3) = 1/2, sin(pi/4) = root 2 / 2, cos(pi/4) = root 2 / 2.
Step 4: sin(7 pi / 12) = (root 3 / 2)(root 2 / 2) + (1/2)(root 2 / 2) = root 6 / 4 + root 2 / 4 = (root 6 + root 2) / 4.
Step 5: (root 6 + root 2) / 4 is about (2.449 + 1.414) / 4 = 0.9659, which equals sin(105 degrees). It works.
The answer is sin(7 pi / 12) = (root 6 + root 2) / 4.
Tips and common mistakes
- For cos(a + b) = cos a cos b - sin a sin b, the sign in the middle is the OPPOSITE of the sign on the left.
- For sin(a +/- b), the middle sign is the SAME as the sign on the left.
- Split strange angles into pieces like pi/6, pi/4, and pi/3 (30, 45, and 60 degrees).
- tan(a + b) only works if the bottom, 1 - tan a tan b, is not zero.
- These identities lead to sum-to-product and product-to-sum formulas, which help simplify products of sines and cosines.
- Check your exact answer with a decimal on your calculator.
- The difference formulas are the sum formulas with b replaced by -b.
- These identities are used to prove the double-angle and half-angle identities.